Question

A finite conducting wire XY carries a current $I$. The magnetic field $d\overrightarrow{B}$ due to infinitesimal element $d\overrightarrow{l}$ of the conducting wire at a point $P$ as shown in figure is:

• A. $d\overrightarrow{B} = \frac{\mu_0}{4\pi}i^2(\frac{d\overrightarrow{l}\times\overrightarrow{r}}{r})$
• B. $d\overrightarrow{B} = \frac{\mu_0}{4\pi}i(\frac{d\overrightarrow{l}\times\overrightarrow{r}}{r^3})$
• C. $d\overrightarrow{B} = \frac{\mu_0}{4\pi}i(\frac{d\overrightarrow{l}\times\overrightarrow{r}}{r})$
• D. $d\overrightarrow{B} = \frac{\mu_0}{4\pi}i^2(\frac{d\overrightarrow{l}\times\overrightarrow{r}}{r^2})$

Answer 1

Answer: (B)

(2)

Answer 2

The magnetic field $d\overrightarrow{B}$ produced by an infinitesimal current element $I d\overrightarrow{l}$ at a point $P$ located at a position vector $\overrightarrow{r}$ relative to the element is given by the Biot-Savart Law: $d\overrightarrow{B} = \frac{\mu_0}{4\pi} \frac{I d\overrightarrow{l} \times \overrightarrow{r}}{r^3}$ where $\mu_0$ is the permeability of free space, $I$ is the current, $d\overrightarrow{l}$ is the vector representing the infinitesimal length of the wire element, $\overrightarrow{r}$ is the position vector from the wire element to the point $P$, and $r = |\overrightarrow{r}|$ is the distance from the element to the point.

In the given question, the current is denoted by $I$ in the text, but by $i$ in the options. Assuming $i$ in the options represents the current $I$, the Biot-Savart Law becomes: $d\overrightarrow{B} = \frac{\mu_0}{4\pi} \frac{i d\overrightarrow{l} \times \overrightarrow{r}}{r^3}$

Comparing this formula with the given options: (1) $d\overrightarrow{B} = \frac{\mu_0}{4\pi}i^2(\frac{d\overrightarrow{l}\times\overrightarrow{r}}{r})$: Incorrect. The current dependence is $i^2$ instead of $i$, and the denominator is $r$ instead of $r^3$. (2) $d\overrightarrow{B} = \frac{\mu_0}{4\pi}i(\frac{d\overrightarrow{l}\times\overrightarrow{r}}{r^3})$: Correct. This matches the Biot-Savart Law. (3) $d\overrightarrow{B} = \frac{\mu_0}{4\pi}i(\frac{d\overrightarrow{l}\times\overrightarrow{r}}{r})$: Incorrect. The denominator is $r$ instead of $r^3$. (4) $d\overrightarrow{B} = \frac{\mu_0}{4\pi}i^2(\frac{d\overrightarrow{l}\times\overrightarrow{r}}{r^2})$: Incorrect. The current dependence is $i^2$ instead of $i$, and the denominator is $r^2$ instead of $r^3$.

Therefore, option (2) is the correct expression for the magnetic field $d\overrightarrow{B}$.